Determine whether the span of one set of vectors contains the span of another set of vectors How can I determine whether the span of a set of vectors (such as $\mathrm{span}\{(3, 1), (4,1), (0,1)\}$ contains the span of another set of vector?
EDIT: I realize that my original question was too vague. If A and B are sets of vectors in $\mathbb{R}^3$, how can you determine whether $\mathrm{span}\{A\}$ contains $\mathrm{span}\{B\}$?
 A: If $M(A)$ is a matrix with the vectors of $A$ as columns and $M(AB)$ is the matrix with the vectors of both $A$ and $B$ as columns, then $span(B) \subset span(A)$ if $rank(M(A))=rank(M(AB))$. Rank is after all the dimension of the column space of a matrix.
A: The simple case is where $\text{span}(A)$ contains three linearly independent vectors in $\mathbb{R}^{3}$, which of course will form a basis set for the vector space (and therefore all vectors in $\mathbb{R}^{3}$ are contained within that spanning set).
In this case it would simply be sufficient to demonstrate that 3 of the vectors within the $\text{span}(A)$ are linearly independent.
If this is not the case, then it suffices to simply show that each of the linearly independent vectors of $\text{span}(B)$ is linearly dependent upon some set of vectors $S \in \text{span}(A)$. Of course if there are more linearly independent vectors in $\text{span}(B)$ then it is impossible for $\text{span}(A)$ to contain $\text{span}(B)$.
A: You need to show that each vector in $B$ (or, if you prefer, a set of vectors whose span contains $B$) can be spanned by vectors in $A$. Then, given a vector in the span of $B$, you can write it as a linear combination of vectors in $B$, each of which can in turn be written as a linear combination of vectors in $A$, so that any vector in the span of $B$ is thus in the span of $A$.
Note that this sufficient condition, that every vector in $B$ is in the span of $A$, is also clearly necessary, since the span of $B$ contains each vector in $B$.
